Question 810: Electrostatics & Capacitance

Question

Three concentric metallic spherical shells of radii $$R,2R,3R$$ are given charges $$Q_1,Q_2,Q_3$$, respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, $$Q_1:Q_2:Q_3$$, is

Accepted Answer

Surface charge density is
$$\rho  = {Q \over {4\pi {R^2}}}$$
where R is the radius of the sphere. For cancelling the field inside the cavity due to the shell of radius R, the shell with radius 2R induces a charge $$-$$Q in its inner surface and hence the total charge on the outer surface is $$Q_1+Q_2$$. Similarly, the charge on the outermost shell is
$${Q_1} + {Q_2} + {Q_3}$$
It is given that the surface charge densities $$\rho_1,\rho_2$$ and $$\rho_3$$ are equal, that is,
$${{{Q_1}} \over {4\pi {R^2}}} = {{{Q_1} + {Q_2}} \over {4\pi {{(2R)}^2}}} = {{{Q_1} + {Q_2} + {Q_3}} \over {4\pi {{(3R)}^2}}}$$
$$ \Rightarrow {Q_1} = {{{Q_1} + {Q_2}} \over 4} = {{{Q_1} + {Q_2} + {Q_3}} \over 9}$$
That is,
$${Q_1} = {{{Q_1}} \over 4} + {{{Q_2}} \over 4}$$
$$ \Rightarrow {Q_2} = 3{Q_1}$$
Therefore,
$${Q_3} = 5{Q_1}$$
Hence, the ratio of the charges given to the shells $${Q_1}:{Q_2}:{Q_3}$$ is 1 : 3 : 5.

Answer

1 : 2 : 3

Answer

1 : 3 : 5

Answer

1 : 4 : 9

Answer

1 : 8 : 18

Electrostatics & Capacitance

Three concentric metallic spherical shells of radii $R,2R,3R are given charges Q_1,Q_2,Q_3, respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, Q_1:Q_2:Q_3$, is

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